Multiple shooting with neural differential equations

Neural differential equations have recently emerged as a flexible data-driven/hybrid approach to model time-series data. This work experimentally demonstrates that if the data contains oscillations, then standard fitting of a neural differential equation may give flattened out trajectory that fails to describe the data. We then introduce the multiple shooting method and present successful demonstrations of this method for the fitting of a neural differential equation to two datasets (synthetic and experimental) that the standard approach fails to fit. Constraints introduced by multiple shooting can be satisfied using a penalty or augmented Lagrangian method.